People asking what's wrong with saying "it won't bite" - I obviously can't answer for anyone else on this thread but in the case of my DD, I don't think she's scared of being bitten - I don't know if she even knows that's a possibility.
She is, however, frightened of being knocked over, which has happened on several occasions, and generally scared of a large (in proportion to her) unpredictable animal getting into her space.
She has various SN, which aren't apparent unless you interact with her (rather than calling from a distance), she's very small for her age and her balance is poor, so being knocked over is a very real possibility.
She's not that bothered whether the dog is friendly or ravenous, because either way the end result tends to be her ending up on her bum on the path! And a dog owner calling "don't worry, he won't hurt you" as she sits wailing in a puddle is a bit annoying.
And we are working on it. Neither DH or I are scared of dogs (neither of us a particular fans, but not scared), so we haven't passed it on. I have had (once) a dog owner tell me off for 'encouraging' her fear, because I picked her up to get above a jumpy dog which had run up to us. I just don't think the uncontrolled interactions when dogs off the lead run up to us are the time to work on her fears - I see those as times to show DD I always have her back and will keep her safe. Dog owners might see that as an over-precious reaction, I don't know. But it annoys me when people assume I'm not working on this with her, because they can't see it in that moment.
When we're walking and we meet a dog she'd bothered by (she is getting much better, doesn't tend to worry if they're on a lead these days), we stand still, let the dog pass us, and I will say something like "look, he's not interested in you at all!' as they go past, or point out what a lovely colour the dog is, or similar positive comments. Her big brother will happily interact with any dog we see after asking the owner, so she sees positive interaction there, too.
If we're walking along a path and a dog comes up behind us which is worrying DD, I'd probably pause and let the dog and owner get past, which seems sensible to me, and as far as I can see is what the OP describes, so I'm not sure now whether that's reasonable or not.
Blimey that was long. Sorry.